Saturday, December 28, 2013

Before we start, I'd like to thank listeners sblack4, WalkerTxClocker, and LaxRef for some more nice reviews on iTunes. Thanks guys!

Now, on to today's topic. During this holiday season in the U.S., many of you are spending time with your families, and taking pity on those who don't have many relatives to visit. (Or taking pity on those with too many relatives visiting.) Sometimes this might bring to mind the strange story of one mathematician who chose never to marry or have children, instead devoting his entire life to his mathematics: Paul Erdos, as profiled in Paul Hoffman's famous biography "The Man Who Loved Only Numbers". Erdos was a Hungarian Jew born in Budapest in 1913. He originally left his country out of fear of anti-Semitism soon after receiving his doctorate, and then spent the rest of his life, until his death in 1996, traveling from university to university taking various temporary and guest positions.

There are many surprising and contradictory aspects to Erdos's life. You would think that someone who chose not to start a family or even to settle in one place would be some kind of social recluse, but Erdos was the opposite. He considered mathematics to be a social activity, not the domain of isolated geniuses behind closed doors. His travels were constantly motivated by the desire to collaborate with other mathematicians, where he would help them solve particularly tough problems. Though he wasn't the leader in any single field of mathematics, never winning the Fields Medal for example, he co-authored about 1525 papers in his lifetime, with 511 different co-authors. Despite being very odd, and sometimes coming across like a homeless drug addict due to his lack of social graces, he was very popular and well-liked in the mathematical community.

Due to his large number of co-authors, the concept of an 'Erdos Number' became a common in-joke in the math world. If you wrote a paper with Paul Erdos, your number was 1. If you wrote a paper with a co-author of his, your number was 2, and so on. It is said that nearly every practicing mathematician in the world has an Erdos number of 8 or less. Incidentally, I found a cool Microsoft site online (linked in the show notes) to search for collaboration distances between two authors, and found that despite being an engineer rather than a mathematician, I have the fairly respectable Erdos number of 4. Perhaps the most famous person with a low Erdos number is baseball legend Hank Aaron. Since he and Erdos once signed the same baseball, when they were both granted honorary degrees on the same day and thus were sitting next to each other when someone requested an autograph, Aaron's Erdos number is said to be 1.

But as you would expect with someone who constantly travelled and never settled down, Erdos had a rather quirky personality that people who worked with him would need to get used to. He had his own unique vocabulary, for example. Children would be referred to as "epsilons", referencing the Greek letter typically used to refer to infinitesmally small quantities. Women and men were "bosses" and "slaves", while people who got married or divorced were "captured" and "liberated". Perhaps this reflected an internal attitude that negatively impacted his potential for dating, even if he had ever given a thought to such things. If someone retired or otherwise left the field of mathematics, Erdos would refer to them as having "died". At one point he was very sad about the "death" of a teenage protegee, having to clarify to symathetic friends that the cause of death was his discovery of girls. The United States and Soviet Union were "Sam" and "Joe". He referred to God as the "Supreme Fascist" or "SF" for short, apparently in protest at being expected to obey the will of divine beings.

Related to proving mathematical results, Erdos had one piece of private vocabulary that was very important to him. He always imagined that somewhere up in the heavens, God had a Book in which were listed the most elegant and direct proofs for every conceivable mathematical result. He wasn't sure if he believed in God, but he definitely believed in the Book. So if he heard a solution to a problem that he liked, he would always say, "That's one from the Book". And if he heard a proof that was valid but seemed very awkward or roundabout, he would acknowledge its validity, but still want to search for the one in the Book. A classic example of a non-Book proof might be Andrew Wiles's famous proof of Fermat's Last Theorem: while it was fully valid and a work of genius, it took hundreds of pages and is understood by very few people in the world. Many still hope that a more elegant solution is out there somewhere, waiting to be found.

Despite Erdos's genius and his sociability, there were many aspects of modern life that either baffled him, or were simply considered beneath his notice, and as a result he constantly depended on his many friends to help him get by. He didn't learn to butter his bread until the age of 21, for example, and always needed help tying his shoes. If left alone in a public place, he would panic and have a lot of difficulty finding his way back to his university or hotel room. If he suddenly thought of a solution to a problem he had been working on, he would call his colleagues at any hour of the day or night, with no consideration for whether it might be a convenient time. He didn't like owning anything, travelling with a single suitcase and requiring his hosts to wash his clothes several times per week. (It always had to be his hosts doing the washing, since he never bothered to learn how to use a washing machine.) The last novel Erdos read was in the 1940s, and he did not watch movies since the 1950s.

On the positive side, his lack of concern for money made him quite generous to fellow mathematicians and others in need. When he won the $50000 Wolf Prize, he used most of the money to establish a scholarship fund in Israel. He would sometimes give small loans of $1000 to struggling students with strong potential in math, telling them to pay him back whenever they had the money. At one point he was seen to take pity on a homeless man just after cashing his paycheck: he took a few dollars out of the envelope to meet his own needs, then handed the rest of the envelope to the stunned beggar. In addition, Erdos would put out "contracts" on math problems he wanted help solving, ranging from $10 to $3000 depending on his estmates of the difficulty. Some of his friends pledged to continue honoring the contracts after his death; at the time, it was estimated that he had about $15000 in contracts still outstanding.

Anyway, this short summary just touches on a few of the bizarre personality quirks in the unusual life of Paul Erdos. If you are as intrigued as I was, be sure to check out Hoffman's biography, "The Man Who Loved Only Numbers". And may the Supreme Fascist grant you a happy new year.

Sunday, December 8, 2013

Recently the phrase "squaring the circle" seems to have been popping up more often than usual, perhaps in response to certain problems currently faced by the U.S. government. I could easily do a very long podcast on some of those topics, but you're probably here to talk about math, not politics. So let's get back to that phrase, "squaring the circle", which means to solve an impossible problem. It descends from a problem originating from ancient times: with no tools except a compass and a straightedge, can we construct a square with the same area as a given circle? Mathematicians struggled with this problem for thousands of years, until it was definitively proven impossible in 1882.

Before we discuss why this is impossible, let's review what the problem is. We start out with a circle of a known radius, drawn on a piece of paper. For simplicity in this discussion let's assume that the radius is 1, with the radius itself defining our unit of measurement. We are allowed to use a compass, which lets us draw a circle around any point with a given radius, and a straightedge, which allows us to draw a line connecting any two points. By the way, no cheating and using markings on the straightedge for measuring distances. Using just these tools, and without being allowed to do things like create new lines of precise lengths, we want to draw a square on the paper whose area matches that of the circle. In effect, this means that starting from a circus of radius 1, and thus an area equal to pi, we need to construct a square with area pi, and thus with each side equal to the square root of pi.

This is one of the most ancient problems known to mathematics, first appearing in some form in the Rhind Papyrus, found in ancient Egypt and dating back to 1650 B.C., though the Egyptians were happy with an approximate solution, treating pi as 256/81 rather than its real irrational value. The Greeks who followed them, starting with Anaxagoras in the 5th century B.C., had a more sophisticated understanding of mathematics, and were the first to insist on searching for a fully accurate solution rather than an approximation. Philosopher Thomas Hobbes inaccurately claimed to have squared the circle in the 1600s, leading to an embarrassing public feud between him and mathematician John Wallis. There were many other discussions and attempts to solve this problem down through the centuries, most amusingly even including two days of frantic scribbling by Abraham Lincoln at one point in the 1850s. It may have been for the best that he gave up math and returned to politics.

Nowadays, the basic concept of why circle-squaring can't be done is not that hard to grasp, with an elementary knowledge of high-school level math. Ultimately it stems from the equivalence between geometry and algebra shown by the field of analytic geometry. View the paper on which you drew the starting circle as a coordinate plane, with every point described by an x and y coordinate. Recall that both lines and circles are described by simple types of equations in such a plane. Lines are described by linear equations of the form y = mx + b, and circles are essentially polynomials in the form x^2 + y^2 = r^2. As a consequence, compass and straightedge constructions are essentially computing a combination of linear and square root functions of your starting lengths. This means that any multi-step construction is building up a set of these linear and square root operations.

But remember the target we were shooting for: we want a square of the same area as our starting circle, so each side of our square measures the square root of pi. However, pi is a transcendental number: this means you cannot get to this value by solving any set of polynomial equations with integer coefficients. Note that this is a stronger condition than being irrational: many irrationals are non-transcendental. The square root of 2, for example, is irrational, but can be derived from the simple equation "x^2 = 2." If a number is transcendental, this implies that no combination of linear and square-root operations, starting with an integer length, could ever reach a value of pi or its square root, and thus our construction of a length pi starting from a circle off radius 1 is impossible. This last part of the argument, the fact that pi is trancendental, was the hardest part of the proof, and the most recent to be put in place, proven in 1882 by German mathematician Ferdinand von Lindemann. While we can create constructions for arbitrarily precise approximations, we can never correctly derive a square with the exact same area as a given circle.

We should also point out that this impossibility is implicitly assuming we are talking about ordinary, Euclidean geometry. If our discussion is including curved spaces, as we discussed in episode 35, all bets are off. These curved spaces have many properties that upend our usual notions of geometry, such as triangles whose angles sum to more or less than 180 degrees. If instead of a flat plane we are sitting on the surface of a curved saddle or a sphere, then we can essentially get a 'free' pi by drawing a line that becomes a curve due to the space's curvature, eliminating a key component of the impossibility proof. I think we can all agree, though, that the ancient Greeks who posed the problem would consider this type of solution to be cheating.

What is most amusing about squaring the circles is that even after its impossibility was very solidly proven in 1882, enthusiastic amateurs kept on trying to "solve" the impossible problem for many years after. This has some analogues in government as well... but I'll try again to keep off that tangent. You may recall that back in episode 20, I talked about an 1897 attempt by a confused Indiana resident to legislate that pi was really equal to 3.2-- this was partially based on his supposedly successful attempts to square a circle using this value. Before you laugh too hard at him or the legislators who actually listened to him, we should note that somehow he got his bogus circle-squaring method published in the American Mathematical Monthly, which will forever be an embarrassment to that publication. In 1911, British professor E.W. Hobson wrote, "Every Scientific Society still receives from time to time communications from the circle squarer and the trisector of angles, who often make amusing attempts to disguise the real character of their essays... The statement is not infrequently accompanied with directions as to the forwarding of any prize of which the writer may be found worthy by the University or Scientific Society addressed, and usually indicates no lack of confidence that the bestowal of such a prize has been amply deserved as the fit reward for the final solution of a problem which has baffled the efforts of a great multitude of predecessors in all ages. " The circle squaring craze seems to have mostly died off in the latter part of the 20th century, but a 2003 article by NPR's "Math Guy" Keith Devlin mentioned that he still regularly received crackpot letters from circle-squarers. And even today, if you look up squaring the circle on Yahoo Answers, you'll find a post by some misguided soul who refuses to believe that the problem is unsolvable, stating that " Its possible we lack the mathematical and conceptual understanding to construct it." I guess anything is possible, but if you trust modern mathematics at all, it's pretty clear that the circle will never be squared.

Sunday, October 13, 2013

If you're a fan of this podcast, I'm betting that at some point in your life you had a picture by M.C. Escher up on your wall, probably before you got married. As you may recall, M. C. Escher was the 20th-century Dutch artist famous for his many woodcuts and lithographs that used geometrical tricks to depict objects that could not possibly exist in real life. While Escher didn't have any formal mathematical training, it is said that he was inspired by a trip to the Alhambra in 1936, and its unique use of many of the 17 possible "wallpaper group" symmetries in its designs. After this, he studied an academic paper by mathematician George Polya on plane symmetries, and started drawing different types of geometric grids to use as the basis of his art. He later studied works of other mathematicians, basing lithographs on mathematical concepts like representations of infinity and hyperbolic plane tilings. As a result, Escher's art has been continually popular among scientists, engineers, and mathematicians.

When you look at one of Escher's impossible illustrations, it's always amusing to think about what it would take for it to exist in real life. But an Israeli professor, Gershon Elber, has gone one step further, and actually created physical models and 3-D printable CAD files, that allow physical creation of impossible Escher objects. How can you create an impossible object, you might ask? The key is the power of optical illusion. Each of these objects only looks correct, in other words matching the original Escher picture, from certain viewing angles-- if you rotate it or look from the wrong direction, you'll see that it is seriously distorted. At this point, you might say that Elber cheated, but give him a break-- we're talking about truly impossible objects here. If you learn how to fully warp spacetime at some point and make them non-impossible, then you can freely scoff at him.

One simple example is what's known as a 'Necker cube'. This simple illusion is based on the pseudo-3-D line drawing of a cube that we all learn in elementary school: basically two squares connected by diagonals on the corners. The impossible part comes when you give the lines some thickness, and draw the cube so that when edges in the drawing meet, they cross each other in contradictory ways, forcing the 'far' corner to be closer than the 'near' corner in some instances. If you haven't seen this before, you can find it at the links in the show notes. In Elber's version, it does indeed look like this miraculous cube has been created in real life-- if viewed from exactly the right angle. If you rotate it a little, you'll see that the whole thing is a mess of curved and distorted pieces, nothing like a real cube.

But Elber doesn't only tackle simple geometrical shapes-- he also recreates complete Escher artworks, such as the "Belvedere", a famous depiction of an impossible two-story tower. The two floors appear to be right on top of each other, a fact required by the placement of several supporting pillars. Yet when you look more closely, the picture also requires the two floors to exist at perpendicular angles; if one is going north-south, the other must be going east-west. Since the floors can't be simulatenously right on top of each other and at perpendicular angles, it's a truly impossible object. Again, you can see the picture in the show notes if you don't recognize it from my description. And once again, what looks like a perfect 3-D construction from a certain viewing angle is a distorted mess of angled and curved support beams from any other angle. For it to look like Escher's real Belvedere, you have to be at the exact spot where the angled beams will look straight to you.

By the way, the urge for a real-life Belvedere is apparently not unique to Elber: I also found a link where another self-described online "professional nerd" named Andrew Lipson constructed a similar model, entirely out of Legos! Lipson's version is actually more complete, even including the minor details and people in Escher's picture. He acknowledges that he needed to cheat at certain points-- for example, quote, "In Escher's original, he's holding an 'impossible cube', but in our version he's holding an impossible LEGO square. Well, OK, not quite impossible if you've got a decent pair of pliers (ouch)."

On Elber's and Lipson's web pages you can find a number of other Escher re-creations. Now while it's pretty fun to say you have created an impossible object in real life, one might argue: if it only looks like the truly impossible object from a certain viewing angle, aren't you cheating? Isn't this roughly equivalent to just creating the 2-D painting anyway? To someone who asks that question, I can only answer that they probably lack the geekiness level required to appreciate the coolness of Elber's and Lipson's work. I suspect most fans of this podcast don't have that problem.

Monday, May 27, 2013

In honor of my recent election to the Hillsboro, Oregon School Board, I thought it might be nice to discuss an education-related topic that's been popping up a lot recently. Back in the 90's there was a fad for what became known as "discovery-based" K12 math curricula. This means that rather than teachers directly telling the students how to do standard operations, such as adding n-digit numbers or simplifying equations, the students would be directed in small groups to experiment and discover such ideas themselves, with some guidance from the teacher. These were widely criticized and debunked-- see the "Mathematically Correct" website linked in the show notes-- and largely fell out of favor. But in recent years, connected with the Common Core movement in the U.S., new forms of these type of programs seem to be coming back. It seems like there is an inherent conflict between two approaches to math education: should students be helped to discover the basic principles of math, or should they be presented with and drilled in standard algorithms? There are some basic criticisms of these discovery techniques that have a lot of weight: the readiness of children to derive standard algorithms, and the internalization of math basics.

As a motivating example, I have a link in the show notes to a video of a young girl adding some 3 and 4-digit numbers, using the methods she learned in a "modern" school. She tries two methods. First she draws pictures representing the thousands, hundreds, tens, and ones in each of the numbers, and after a tedious 8 minutes of counting pictures, gets a wrong answer. Then she uses the standard method (which her mom taught her), of writing the numbers on top of each other, adding in columns, and carrying when needed-- and in less than a minute has the correct answer. The whole process of drawing the pictures seems rather absurd, and it's clear that the girl really has no understanding of how the pictures really relate to place-based notation, or of how the standard algorithm is really just an advanced abstraction of her picture-method. Several online commentators pointed out that the picture method was directly analogous to roman numerals, which no sensible person would use today for nontrivial calculations.

The first question to ask is: are children in these classes ready to derive the standard algorithms that took mathematicians thousands of years to create? Going from simple direct methods, like the little girl's pictures which were analogous to roman numerals, to the modern algorithms, requires some clever abstract reasoning. As I discovered when I tried to clarify the higher-dimensional concepts from the Flatland movie to my 6 year old, there are some types of logic and abstraction that a child may not be ready for. Many psychologists talk about Piaget's stages of cognitive development: the "symbolic function" stage at ages 2-4; the "intuitive thought" stage at ages 4-7; the "concrete operational" stage at ages 7-11; and the "formal operational" stage beyond that. You can see more details at the link in the show notes. The exact ages vary from child to child of course, but the key point here is that at the first three stages, a child's ability to develop abstract algorithms from concrete examples is severely limited; and the capacity for true abstract reasoning isn't really developed until high school for a majority of kids. This is probably one reason why the girl in the video had trouble seeing the connection between the picture-based method and the standard method of addition. But at these younger ages, kids are much better at picking up and internalizing rote facts and procedures, which would seem to make it an ideal time to teach them standard algorithms.

We also need to ask whether these new types of lessons, assuming they focus on both the historical derivation and the current best-known-methods, will actually result in students learning the standard algorithms. Here there is reason for concern. Often the modern math lessons have many fewer practice problems in the homework than traditional math, and encourage the use of calculators and computers to do basic calculations. Unfortunately, these are missing a very imporant aspect of all this drilling-- it helps people to really absorb basic mathematical algorithms, and make them instinctive. Back in episode 70, "Number Nonsense", I discussed my frustration with a fast food cashier who could not recognize that 2+2=4 without a calculator. And during my recent school board campaign, I came across a mom who was upset that her child took out a calculator when asked the difference between 6 percent and 600 percent. More important than these anecdotes, though, is that in order to have any hope of success in advanced science and engineering classes, kids really need this basic number sense. Even if the hard problems are ultimately crunched by computer programs, it's simply not possible to have an initial discussion of an engineering problem if every rough estimate needs a pause to get out a calculator. The little girl in the video was lucky that her mom took it in her own hands to work with her at home and make sure she had a good understanding of the standard algorithm.

Our standard methods, like the one that enabled a young girl to add 4-digit numbers in less than a minute, were developed after thousands of years of thought by very smart people. And since then they have been used regularly by a huge population of businessmen, scientists, and engineers, the majority of whom have never given a thought to the details of how these were derived or why they work. A lot of proponents of the new teaching methods criticize the "drill and kill" of traditional math, the many exercises given to practice and memorize standard algorithms, which they consider boring. But the advantages of all this drilling is that using these techniques becomes easy and instinctive: how often do we stop to ask ourselves why we are able to do simple addition problems? Well, maybe Math Mutation listeners do, but most people can add just fine without thinking about the historical development of place-based notation. Being able to do basic math opens the doors to advanced concepts of science, engineering, and computers. And students who have NOT mastered these basic foundations will find these important topics forever beyond their reach.

So, am I advocating that we throw out all these modern and Common Core math programs, and go back strictly to traditional methods? Not necessarily. A properly designed program which teaches and drills the standard algorithms, while also emphasizing problem-solving and using the history of their derivation to provide some motivation and color to the class, might very well be a great success. But in the urge to eliminate the drill-and-kill method and make math more fun, we seem to be constantly rediscovering the lesson that Euclid taught to Ptolemy over 2000 years ago, that there is no "royal road" to mathematics. We need to be very careful of the tendency of education efforts every few decades to come up with silver bullets that will somehow make math easy for everyone-- internalizing the basics algorithms of math usually requires a lot of concentrated thought and hard work. We've been here before, as you might recall: back in episode 145 I discussed the New Math movement of the 1970s, similarly based on professors promising that adding theoretical foundations to elementary mathematics would somehow make it easier to learn, and ending in disaster. Many aspects of math really can be fun, as I hope most of my podcast episodes have shown, but to understand the subject overall you really need a solid mastery of the basics.

By the way-- I know from listener emails that many of you out there are actually math or science teachers. I'd really like to hear from you about experiences, positive or negative, with these new mathematics programs. Please send emails to erik@mathmutation.com. Thanks!

Monday, April 29, 2013

Before we begin, I'd like to thank listener Bret McDermitt for posting another nice review on iTunes. I'd also like to thank the Math Insider website for featuring Math Mutation in an article on great math podcasts-- check the article out at the link in the show notes! Now, let's get on to today's topic.

As I looked out the window recently on a windy day, watching leaves and dirt fly around outside, it brought to mind Rudy Rucker's concept of a 'paracomputer', a computer based on observing natural processes that happen to be equivalent to a natural computation. Let's review the background of this idea briefly so we can explain it.

You may recall from some earlier podcasts the ideas of Stephen Wolfram's controversial tome "A New Kind of Science", which claims that we can view many natural phenomena as similar to a computation such as a cellular automaton. One popular example of a cellular automaton is Conway's Game of Life. In this game, each square in a grid can be "alive" or "dead", usually represented by being colored black or white. A live square stays alive for the next time step if and only if it has exactly two or three live neighbors, otherwise dying of loneliness or overcrowding. And a dead square is "born" and becomes alive if it has exactly three live neighbors. The amazing thing about this simple game is that depending on the initial set of live squares, it can display a huge range of lifelike behaviors, as you can see illustrated on the Wikipedia page linked in the show notes. This led researchers such as Wolfram to hypothesize that such simple computations might explain the basics of how our universe really works.

If you look at the patterns created by various Life configurations, you can see three basic types. Predictable patterns settle into an endless repetition of simple elements, such as a square group of cells simply living forever in the middle. Chaotic patterns end up looking totally random, like the static on a malfunctioning TV set. The most interesting ones are what Rudy Rucker calls "gnarly" patterns, which seem to have interesting structure but are unpredictable. These gnarly patterns are what seem to have promise to represent real life: for example, a gnarly pattern might look analogous to clouds moving in the sky, the life cycle of a jellyfish, or celestial bodies interacting over the lifetime of a solar system. These all share the property that their ultimate results are in theory predictable, but so complex to calculate that there's no fundamentally better way to figure out what will happen than to wait and see.

After observing many of these gnarly systems, Wolfram hypothesized the Principle of Computational Equivalence. This states that at some fundamental level, all these gnarly real-life systems are each capable of acting as a universal computer, and thus equivalent to each other. So, for example, by coming up with a suitable "mapping" of clouds that represents the initial state of a jellyfish being born, and somehow manipulating the clouds into this configuration, you would be able to, by watching those clouds, predict exactly what is happening at any stage of the jellyfish's life. You just need to decode the current state of the clouds into the equivalent state of the jellyfish.

And this is Rudy Rucker's idea of a "paracomputer". The concept is that you figure out some gnarly physical phenomenon that you can control, identify the proper mapping to and from whatever other type of system you want to analyze, and just set it in motion. Sounds pretty cool, right? So, all those old fortune-tellers who claim to be able to tell the future from tea leaves might actually have the core of a feasible idea: all you have to do is figure out enough detail about your life to map its precise state into the tea leaves in a very large kettle, shake the kettle in a defined way so the leaves will represent each future state of your life, read the results in the leaves, and map the final leaf pattern to the details of your life. Presto! You now know every detail of your future.

Now if you're picky, you might have spotted a few flaws in this idea. Wolfram's Principle itself is very controversial, as it seems to collapse what could potentially be many different complexity classes, and claim that a huge range of phenomena are equally complex to compute. His observations provide some evidence, but nothing close to a real proof. There's also the question of how you would fully figure out the starting state of your life, to map to the tea leaves: you might object as the fortune-teller takes out a bone saw, opens up your skull, and carefully extracts the state of each neuron in your brain. Even after your brain has been carefully put back together without modifying the state, then there's the question of how many tea leaves it would take to map your life: the fortune-teller might need all the tea in China to properly represent it, especially if you live the exciting life of a math podcast fan. And finally, there's the question of how long it would take the tea leaves to reach their representation of any point in your future: would this really be any faster than just waiting for your future to happen?

But, nevertheless, it is kind of fun to think that nature really consists of computers all around us, just waiting to be properly understood. Maybe soon you'll be able to turn in your PlayStation 3 and play Call Of Duty with leaves blowing in the wind in your backyard.

Sunday, January 6, 2013

Before we start I'd like to thank listener SkepticHunter for posting another nice review on iTunes. Don't forget to post one too if you also like the podcast!

Recently I installed Google Earth on my iphone, and showed my daughter the cool feature where you can start with a view of the whole planet, and then zoom in to the exact spot where you currently are. Playing with this feature brought to mind an amusing essay I had recently read in Umberto Eco's "How To Travel With A Salmon", called "On The Impossibility of Drawing a Map of the Empire on a Scale of 1 to 1". The essay discusses a project of creating a map that is so detailed, it actually has a 1:1 scale, with each element on the map being the same size as the feature described.

The silly idea of a 1:1 scale map was apparently first proposed by Lewis Caroll in his lesser-known book Sylvie and Bruno, a somewhat enjoyable tale that didn't quite have the narrative flow of Alice in Wonderland, but shared its sense of absurdity. At one point, a character brags about his kingdom's mapmaking skills:What do you consider the largest map that would be really useful?"
"About six inches to the mile."
"Only six inches!" exclaimed Mein Herr. "We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile!"
"Have you used it much?" I enquired.
"It has never been spread out, yet," said Mein Herr: "the farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well."I think that quote pretty well speaks for itself.

Argentine writer Jorge Luis Borges was apparently a fan of Carroll's, and a few decades later in a short-short story commented on the idea himself. In Borges's vision, the 1:1 scale map is actually constructed, but then abandoned as useless, and left in western deserts to be gradually eroded away by the weather. In another of his essays, Borges points out that if England contains a perfect map of England, then that map must contain a depiction of the map itself, since that map is part of England-- and that depiction must include the depiction of the map, etc, out to infinity.

Eco's essay goes into much more detail than Carroll or Borges, discussing the major requirements for a 1:1 map to be useful. He insists that the map be able to exist within the country being mapped; I guess this is because it's not useful as a travel reference if the map must be elsewhere. It must be an actual map, rather than a mechanically created replica: for example, it would be cheating to take a plaster cast of the whole country and call it a map. The map also needs to be useful as a tool for referencing other parts of the country: so it can't be the case that for any location X, the depiction of X on the map lies on the portion of the map directly located at X, since then the map would be no more useful than a transparent sheet over the land. He also discusses various physical constraints such as the materials needed, folding techniques, and the effect on the actual country of having a giant map rolled out over it.

I also found an online discussion that pointed to several lesser-known references to such 1:1 scale maps. The rock band They Might Be Giants sings about a ship that is a 1:1 scale map of the state of Arkansas. The British comedy series "Blackadder" has an episode where an incompetent general is constructing a tabletop replica of recently conquered territory on a 1:1 scale. And modern comedian Steven Wright has a joke that goes "I have a map of the United States...actual size. It says, 'Scale: 1 mile = 1 mile.' I spent last summer folding it."

This whole discussion also seems to relate to the famous philosophical statement by Alfred Korzybski, "The map is not the territory". He was pointing out that if you are referring to a map, alternate view, or even a mental abstraction of something, you shouldn't confuse that with the thing itself. Naturally, this doesn't only refer to literal maps, but to just about anything you might see, interact with, or think about. This leads to another infinite regression problem, as we cannot actually sense physical things directly: we are always reacting to images on our retina, models in our mind, sets of beliefs we have about reality, or similar abstractions. So even when we think we are directly interacting with actual reality, we are actually dealing with somewhat less fidelity than the ideal 1:1 map.

But, in any case, what I consider the most ironic thing about all these riffs on Lewis Carroll's original attempt at absurdity is that the whole concept of 1:1 scale maps is no longer so absurd. Modern software tools like Google Earth really do allow us to depict arbitrary scales, even 1:1, by storing the map virtually and letting us just zoom in on the parts we currently need. Technically I don't think we have quite the satellite resolution to consider Google Earth maps to be 1:1 scale, but we're getting pretty close. And I'm sure philosophers will continue to point out that the Google Map is not the territory-- but for practical purposes, it seems close enough to me. I wonder what other ridiculous absurdities from Alice in Wonderland or Sylvie and Bruno will be rendered mundane by future technologies.

Note that this podcast is intended mainly for audio consumption, so you will not see the numerous illustrations & diagrams you would find at most math sites, though these are linked in the show notes whenever possible.

The Math Mutation Book

The Math Mutation book, "Math Mutation Classics: Exploring Interesting, Fun, and Weird Corners of Mathematics", is now available. You can order it from Amazon at this link.

By the way, if you like it, don't forget to post a review on Amazon. And of course, if you're in the Portland, Oregon metro area, I'll be happy to autograph your copy sometime.

Donations

If you enjoy this podcast enough to donate money, thanks! But since I don't spend an appreciable amount on creating it, I ask instead that you donate to your favorite charity, and send me an email about it. You will then experience fame and fortune as I mention your name on the next podcast.